| 1. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
- 2010
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 235, s. 563-592
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Tidskriftsartikel (refereegranskat)abstract
- The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called `Greeks', remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the `Greeks' is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated `Greeks'. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated `Greeks'. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.
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| 2. |
- Avelin, Benny, 1984-, et al.
(författare)
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Boundary estimates for solutions to operators of p-Laplace type with lower order terms
- 2011
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 250:1, s. 264-291
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the boundary behavior of solutions to equations of the form∇⋅A(x,∇u)+B(x,∇u)=0, in a domain Ω⊂Rn, assuming that Ω is a δ-Reifenberg flat domain for δ sufficiently small. The function A is assumed to be of p-Laplace character. Concerning B, we assume that |∇ηB(x,η)|⩽c|η|p−2, |B(x,η)|⩽c|η|p−1, for some constant c, and that B(x,η)=|η|p−1B(x,η/|η|), whenever x∈Rn, η∈Rn∖{0}. In particular, we generalize the results proved in J. Lewis et al. (2008) [12] concerning the equation ∇⋅A(x,∇u)=0, to equations including lower order terms.
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| 3. |
- Önskog, Thomas, 1979-, et al.
(författare)
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Weak approximation of obliquely reflected diffusions in time-dependent domains
- 2010
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Ingår i: Journal of Computational Mathematics. - : Global Science Press. - 0254-9409 .- 1991-7139. ; 28:5, s. 579-605
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Tidskriftsartikel (refereegranskat)abstract
- In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in R^d, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to 1/2 and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.
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| 4. |
- Önskog, Thomas, 1979-, et al.
(författare)
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The Skorohod oblique reflection problem in time-dependent domains
- 2010
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Ingår i: Annals of Probability. - : Project Euclid. - 0091-1798 .- 2168-894X. ; 38:6, s. 2170-2223
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Tidskriftsartikel (refereegranskat)abstract
- The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka and further investigated by P.-L. Lions and A.-S. Sznitman in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of càdlàg solutions (x, λ) to the Skorohod problem, with oblique reflection, for (D,,w) assuming, in particular, that D is a time-dependent domain (Theorem 1.2). In addition, we prove that if w is continuous, then x is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.
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| 5. |
- Frentz, Marie, 1980-, et al.
(författare)
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Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type
- 2010
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 234:1, s. 146-164
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Tidskriftsartikel (refereegranskat)abstract
- Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.
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| 6. |
- Frentz, Marie, 1980-, et al.
(författare)
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The obstacle problem for parabolic non-divergence form operators of Hörmander type
- 2012
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Ingår i: Journal of Differential Equations. - : Elsevier. - 0022-0396 .- 1090-2732. ; 252:9, s. 5002-2041
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we establish the existence and uniqueness of strong solutions to the obstacle problem for a class of parabolic sub-elliptic operators in non-divergence form structured on a set of smooth vector fields in Rn, X={X1,…,Xq}X={X1,…,Xq}, q⩽n, satisfying Hörmanderʼs finite rank condition. We furthermore prove that any strong solution belongs to a suitable class of Hölder continuous functions. As part of our argument, and this is of independent interest, we prove a Sobolev type embedding theorem, as well as certain a priori interior estimates, valid in the context of Sobolev spaces defined in terms of the system of vector fields.
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| 7. |
- Nyström, Kaj, 1969-, et al.
(författare)
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On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains
- 2009
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Ingår i: Monte Carlo Methods and Applications. - Berlin New York : de Gruyter. - 1569-3961. ; 15, s. 11-47
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Tidskriftsartikel (refereegranskat)abstract
- Dirichlet problems for second order parabolic operators in space-time domains Ω⊂ Rn+1 are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [4] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [3] for weak approximation of stochastic differential equations.
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| 8. |
- Lundström, Niklas L.P. 1980-, et al.
(författare)
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On a two-phase free boundary condition for p-harmonic measures
- 2009
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Ingår i: Manuscripta mathematica. - : Springer. - 0025-2611 .- 1432-1785. ; 129:2, s. 231-249
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Tidskriftsartikel (refereegranskat)abstract
- Let Ωi⊂Rn,i∈{1,2} , be two (δ, r 0)-Reifenberg flat domains, for some 0<δ<δ^ and r 0 > 0, assume Ω1∩Ω2=∅ and that, for some w∈Rn and some 0 < r, w∈∂Ω1∩∂Ω2,∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . Let p, 1 < p < ∞, be given and let u i , i∈{1,2} , denote a non-negative p-harmonic function in Ω i , assume that u i , i∈{1,2}, is continuous in Ω¯i∩B(w,2r) and that u i = 0 on ∂Ωi∩B(w,2r) . Extend u i to B(w, 2r) by defining ui≡0 on B(w,2r)∖Ωi. Then there exists a unique finite positive Borel measure μ i , i∈{1,2} , on R n , with support in ∂Ωi∩B(w,2r) , such that if ϕ∈C∞0(B(w,2r)) , then∫Rn|∇ui|p−2⟨∇ui,∇ϕ⟩dx=−∫Rnϕdμi.Let Δ(w,2r)=∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . The main result proved in this paper is the following. Assume that μ 2 is absolutely continuous with respect to μ 1 on Δ(w, 2r), d μ 2 = kd μ 1 for μ 1-almost every point in Δ(w, 2r) and that logk∈VMO(Δ(w,r),μ1) . Then there exists δ~=δ~(p,n)>0 , δ~<δ^ , such that if δ≤δ~ , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).
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| 9. |
- Kuusi, Tuomo, et al.
(författare)
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A Boundary Harnack Inequality for Singular Equations of p-Parabolic Type
- 2014
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 142:8, s. 2705-2719
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Tidskriftsartikel (refereegranskat)abstract
- We prove a boundary Harnack type inequality for nonnegative solutions to singular equations of p-parabolic type, 2n/(n+1)1,1-regular. Simple examples show, using the corresponding estimates valid for the heat equation as a point of reference, that this type of inequality cannot, in general, be expected to hold in the degenerate case ( 2
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| 10. |
- Kuusi, Tuomo, et al.
(författare)
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Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets
- 2014
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Ingår i: Journal des Mathématiques Pures et Appliquées. - : Elsevier BV. - 0021-7824 .- 1776-3371. ; 101:2, s. 119-151
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we prove, by showing that solutions have exactly the same degree of regularity as the obstacle, optimal regularity results for obstacle problems involving evolutionary p-Laplace type operators. A main ingredient, of independent interest, is a new intrinsic interpolative geometry allowing for optimal linearization principles via blow-up analysis at contact points. This also opens the way to the proof of a removability theorem for solutions to evolutionary p-Laplace type equations. A basic feature of the paper is that no differentiability in time is assumed on the obstacle; this is in line with the corresponding linear results.
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